#1409135

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,3)(6,7)(9,10,13,15)(11,12,14,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30);(9,11)(10,12)(13,14)(15,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32);(1,2,4,3)(5,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(5,6,8,7)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32);(1,3,4,2)(5,7,8,6)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1316331);